Nonlocal general vector nonlinear Schroedinger equations:Integrability, PT symmetribility, and solutions

TL;DR

This paper introduces new nonlocal vector nonlinear Schrödinger equations, explores their integrability, PT symmetry, and solutions, and establishes connections with known models, providing exact soliton solutions and conservation laws.

Contribution

It presents a novel family of nonlocal vector NLS equations with integrability and PT symmetry, including exact solutions and conservation laws, expanding the understanding of nonlocal nonlinear wave models.

Findings

Nonlocal G_{-1}^{(nm)} model has Lax pair and infinite conservation laws.

Exact bright, dark, and mixed soliton solutions are found.

The models exhibit PT symmetry and invariance under PT transformations.

Abstract

A family of new one-parameter (\epsilon_x=\pm 1) nonlinear wave models (called G_{\epsilon_x}^{(nm)} model) is presented, including both the local (\epsilon_x=1) and new integrable nonlocal $(\epsilon_x=-1)$ general vector nonlinear Schr\"odinger (VNLS) equations with the self-phase, cross-phase, and multi-wave mixing modulations. The nonlocal G_{-1}^{(nm)} model is shown to possess the Lax pair and infinite number of conservation laws for $m=1$. We also establish a connection between the G_{\epsilon_x}^{(nm)} model and some known models. Some symmetric reductions and exact solutions (e.g., bright, dark, and mixed bright-dark solitons) of the representative nonlocal systems are also found. Moreover, we find that the new general two-parameter (\epsilon_x, \epsilon_t) model (called G_{\epsilon_x, \epsilon_t}^{(nm)} model) including the G_{\epsilon_x}^{(nm)} model is invariant under the PT-symmetric transformation and the PT symmetribility of its self-induced potentials is discussed for the distinct two parameters (\epsilon_x, \epsilon_t)=(\pm 1, \pm 1).